Bounding Eigenvalues with Packing Density
Neal Coleman · arXiv (Cornell University) · 2015
We prove a lower bound on the eigenvalues $λ_k$, $k\in\mathbb{N}$, of the Dirichlet Laplacian of a bounded domain $Ω\subset\mathbb{R}^n$ of volume $V$: $$ λ_k \geq C_n\bigg( δ\frac{k}{V}\bigg)^{2/n} $$ where $δ$ is a constant that measures how efficiently $Ω$ can be packed into $\mathbb{R}^n$ and $C_n$ is the constant found in Weyl's law. This generalizes a result of Urakawa in 1984. If $δ^{2/n} > n/(n+2)$, this bound is stronger than the eigenvalue bound proven by Li and Yau in 1983. For example, in the case of convex planar domains, we have for all $k\in\mathbb{N}$, $$ λ_k \geq \frac{2\sqrt{3}πk}{V}. $$